Integral exponential function
The special function defined for real by the equation
The graph of the integral exponential function is illustrated in Fig..
Graphs of the functions , and .
For , the function has an infinite discontinuity at , and the integral exponential function is understood in the sense of the principal value of this integral:
The integral exponential function can be represented by the series
where is the Euler constant.
There is an asymptotic representation:
As a function of the complex variable , the integral exponential function
is a single-valued analytic function in the -plane slit along the positive real semi-axis ; here the value of is chosen such that . The behaviour of close to the slit is described by the limiting relations:
The asymptotic representation in the region is:
The integral exponential function is related to the integral logarithm by the formulas
The differentiation formula is:
The following notations are sometimes used:
|||H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953)|
|||E. Jahnke, F. Emde, "Tables of functions with formulae and curves" , Dover, reprint (1945) (Translated from German)|
|||A. Krazer, W. Franz, "Transzendente Funktionen" , Akademie Verlag (1960)|
|||N.N. Lebedev, "Special functions and their applications" , Prentice-Hall (1965) (Translated from Russian)|
The function is usually called the exponential integral.
Instead of by the series representation, for complex values of ( not positive real) the function can be defined by the integal (as for real ); since the integrand is analytic, the integral is path-independent in .
Formula (1) with replaced by then holds for , and the function defined by (2) (for ) is also known as the modified exponential integral.
Integral exponential function. A.B. Ivanov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Integral_exponential_function&oldid=14925